Quantization and Isotropic Submanifolds

Prequantization, Geometric Quantization, Corrected Geometric Quantization

“…BKS pairing, used by different authors, is examined in three cases: in the presence of two real polarizations, one real and one complex and, at the end, two complex polarizations. The last section is dedicated to the Bohr-Sommerfeld subvarieties with recent developments (see [8]).…”

“…We can consider the triple ( ) with the machinery of geometric quantization. Following[8], we can study the asymptotic expansion of the squared norm of an "isotropic state" in particular conditions. An isotropic state is a family of sections of an Hilbert space associated to the lagrangian submanifold.…”

Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds

“…We say that (X, ∇ X ) satisfies the Bohr-Sommerfeld condition if there exists a non-vanishing smooth section ζ ∈ C ∞ (X, ι * L) satisfying ∇ X ζ = 0 (cf. [11], [13]). Hence (X, ∇ X ) satisfies the Bohr-Sommerfeld condition if and only if the holonomy of (ι * L, ∇ X ) is trivial.…”

“…We note that Bohr-Sommerfeld Lagrangian submanifolds and related asymptotic series expansion formulas are studied in more general settings (cf. [11]).…”